Additional questions and answers will be added, if they really turn out to be frequently asked.

What is the arrow of time?

The past is different from the future. One of the most obvious features of the macroscopic world is irreversibility: heat doesn't flow spontaneously from cold objects to hot ones, we can turn eggs into omelets but not omelets into eggs, ice cubes melt in warm water but glasses of water don't spontaneously give rise to ice cubes. We remember the past, but not the future; we can take actions that affect the future, but not the past (we can't unto our mistakes). We are all born, then age, then die; never the other way around. The distinction between past and future seems to be consistent throughout the observable universe. The arrow of time is simply
that distinction, pointing from past to future.

Why is there such an arrow?

Irreversible processes are summarized by the Second Law of Thermodynamics: the entropy of a closed system will (practically) never decrease into the future. It's a bedrock
foundation of modern physics.

What's "entropy"?

Entropy is a measure of the disorder of a system. A nice organized system, like an unbroken egg
or a neatly-arranged pile of papers, has a low entropy; a disorganized system, like a broken egg or a
scattered mess of papers, has a high entropy. Left to its own devices, entropy goes up as time passes.

But entropy decreases all the time; we can freeze water to make ice cubes, after all. And life evolved.

Not all systems are closed. The Second Law doesn't forbid decreases in entropy in open systems --
by putting in the work, you are able to tidy up your room, decreasing its entropy but still increasing the entropy
of the whole universe (you make noise, burn calories, etc.). Nor is it in any way incompatible with evolution or complexity or any such thing.

Should we be surprised?

The first mystery of the arrow of time is that it's nowhere to be found in the fundamental laws of physics. Those
laws work perfectly well if we run processes backwards in time.
(More rigorously, for every allowed process there exists a time-reversed process that is also allowed, obtained by switching parity and exchanging particles for antiparticles -- the CPT Theorem.) Nevertheless,
the macroscopic world we observe is full of irreversible processes. The puzzle is to reconcile microscopic reversibility with macroscopic irreversibility.

And how do we reconcile them?

The observed macroscopic irreversibility is not a consequence of the fundamental laws of physics, it's a consequence of the particular configuration in which the universe finds itself. In particular, the unusual low-entropy conditions in the very early universe, near the Big Bang. Understanding the arrow of time is a matter of understanding the origin of the universe.

Wasn't this all figured out over a century ago?

Not exactly. In the late 19th century, Boltzmann and Gibbs figured out what entropy really is: it's a measure of the number of individual microscopic states that are macroscopically indistinguishable. An omelet is higher entropy than an egg because there are more ways to re-arrange its atoms while keeping it indisputably an omelet, than there are for the egg. That provides half of the explanation for the Second Law: entropy tends to increase because there are more ways to be high entropy than low entropy. The other half of the question still remains: why was the entropy ever low in the first place?

Is the origin of the Second Law really cosmological? We never talked about the early universe back when I took thermodynamics.

Trust me, it is. (Or trust Richard Feynman, if you don't trust me.) Of course you don't need to appeal to cosmology to use the Second Law, or even to "derive" it under some reasonable-sounding assumptions. However, those reasonable-sounding assumptions are typically not true of the real world. Using only time-symmetric laws of physics, you can't derive time-asymmetric macroscopic behavior (as pointed out in the "reversibility objections" of Lohschmidt and Zermelo back in the time of Boltzmann and Gibbs); every trajectory is precisely as likely as its time-reverse, so there can't be any overall preference for one direction of time over the other. The usual "derivations" of the second law, if taken at face value, could equally well be used to predict that the entropy must be higher in the past -- an inevitable answer, if one has recourse only to reversible dynamics. But the entropy was lower in the past, and to understand that empirical feature of the universe we have to think about cosmology.

Does inflation explain the low entropy of the early universe?

Not by itself, no. To get inflation to start requires even lower-entropy initial conditions than those implied by the conventional Big Bang model. Inflation just makes the problem harder.

Does that mean that inflation is wrong?

No. Inflation is an attractive mechanism for generating primordial cosmological perturbations, and provides a way to dynamically create a huge number of particles from a small region of space. The question is simply, why did inflation ever start? Rather than removing the need for a sensible theory of initial conditions, inflation makes the need even more urgent.

My theory of (brane gasses/loop quantum cosmology/ekpyrosis/Euclidean quantum gravity) provides a very natural and attractive initial condition for the universe. The arrow of time just pops out as a bonus.

I doubt it. We human beings are terrible temporal chauvinists -- it's very hard for us not to treat "initial" conditions differently than "final" conditions. But if the laws of physics are truly reversible, these should be on exactly the same footing -- a requirement that philosopher Huw Price has dubbed the Double Standard Principle. If a set of initial conditions is purportedly "natural," the final conditions should be equally natural. Any theory in which the far past is dramatically different from the far future is violating this principle in one way or another. In "bouncing" cosmologies, the past and future can be similar, but there tends to be a special point in the middle where the entropy is inexplicably low.

What is the entropy of the universe?

We're not precisely sure. We do not understand quantum gravity well enough to write down a general formula for the entropy of a self-gravitating state. On the other hand, we can do well enough. In the early universe, when it was just a homogenous plasma, the entropy was essentially the number of particles -- within our current cosmological horizon, that's about 1088. Once black holes form, they tend to dominate; a single supermassive black hole, such as the one at the center of our galaxy, has an entropy of order 1090, according to Stephen Hawking's famous formula. If you took all of the matter in our observable universe and made one big black hole, the entropy would be about 10120. The entropy of the universe might seem big, but it's nowhere near as big as it could be.

If you don't understand entropy that well, how can you even talk about the arrow of time?

We don't need a rigorous formula to understand that there is a problem, and possibly even to solve it. One thing is for sure about entropy: low-entropy states tend to evolve into higher-entropy ones, not the other way around. So if state A naturally evolves into state B nearly all of the time, but almost never the other way around, it's safe to say that the entropy of B is higher than the entropy of A.

Are black holes the highest-entropy states that exist?

No. Remember that black holes give off Hawking radiation, and thus evaporate; according to the principle just elucidated, the entropy of the thin gruel of radiation into which the black hole evolves must have a higher entropy. This is, in fact, borne out by explicit calculation.

So what does a high-entropy state look like?

Empty space. In a theory like general relativity, where energy and particle number and volume are not conserved, we can always expand space to give rise to more phase space for matter particles, thus allowing the entropy to increase. Note that our actual universe is evolving (under the influence of the cosmological constant) to an increasingly cold, empty state -- exactly as we should expect if such a state were high entropy. The real cosmological puzzle, then, is why our universe ever found itself with so many particles packed into such a tiny volume.

Could the universe just be a statistical fluctuation?

No. This was a suggestion of Boltzmann's and Schuetz's, but it doesn't work in the real world. The idea is that, since the tendency of entropy to increase is statistical rather than absolute, starting from a state of maximal entropy we would (given world enough and time) witness downward fluctuations into lower-entropy states. That's true, but large fluctuations are much less frequent than small fluctuations, and our universe would have to be an enormously large fluctuation. There is no reason, anthropic or otherwise, for the entropy to be as low as it is; we should be much closer to thermal equilibrium if this model were correct. The reductio ad absurdum of this argument leads us to Boltzmann Brains -- random brain-sized fluctuations that stick around just long enough to perceive their own existence before dissolving back into the chaos.

Don't the weak interactions violate time-reversal invariance?

Not exactly; more precisely, it depends on definitions, and the relevant fact is that the weak interactions have nothing to do with the arrow of time. They are not invariant under the T (time reversal) operation of quantum field theory, as has been experimentally verified in the decay of the neutral kaon. (The experiments found CP violation, which by the CPT theorem implies T violation.) But as far as thermodynamics is concerned, it's CPT invariance that matters, not T invariance. For every solution to the equations of motion, there is exactly one time-reversed solution -- it just happens to also involve a parity inversion and an exchange of particles with antiparticles. CP violation cannot explain the Second Law of Thermodynamics.

Doesn't the collapse of the wavefunction in quantum mechanics violate time-reversal invariance?

It certainly appears to, but whether it "really" does depends (sadly) on one's interpretation of quantum mechanics. If you believe something like the Copenhagen interpretation, then yes, there really is a stochastic and irreversible process of wavefunction collapse. Once again, however, it is unclear how this could help explain the arrow of time -- whether or not wavefunctions collapse, we are left without an explanation of why the early universe had such a small entropy. If you believe in something like the Many-Worlds interpretation, then the evolution of the wavefunction is completely unitary and reversible; it just appears to be irreversible, since we don't have access to the entire wavefunction. Rather, we belong in some particular semiclassical history, separated out from other histories by the process of decoherence. In that case, the fact that wavefunctions appear to collapse in one direction of time but not the other is not an explanation for the arrow of time, but in fact a consequence of it. The low-entropy early universe was in something close to a pure state, which enabled countless "branchings" as it evolved into the future.

This sounds like a hard problem. Is there any way the arrow of time can be explained dynamically?

I can think of two ways. One is to impose a boundary condition that enforces one end of time to be low-entropy, whether by fiat or via some higher principle; this is the strategy of Roger Penrose's Weyl Curvature Hypothesis, and arguably that of most flavors of quantum cosmology. The other is to show that reversibilty is violated spontaneously -- even if the laws of physics are time-reversal invariant, the relevant solutions to those laws might not be. However, if there exists a maximal entropy (thermal equilibrium) state, and the universe is eternal, it's hard to see why we aren't in such an equilibrium state -- and that would be static, not constantly evolving. This is why I personally believe that there is no such equilibrium state, and that the universe evolves because it can always evolve. The trick of course, is to implement such a strategy in a well-founded theoretical framework, one in which the particular way in which the universe evolves is by creating regions of post-Big-Bang spacetime such as the one in which we find ourselves.